Linear programming for the 0–1 quadratic knapsack problem

Billionnet, Alain; Calmels, Frédéric

doi:10.1016/0377-2217(94)00229-0

Cited by 80 publications

(73 citation statements)

References 14 publications

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“…In order to derive a good initial solution, we implemented the heuristic algorithm for QKP devised by Billionnet and Calmels [28], and then used by Caprara et al [17]. This algorithm is based on the greedy principle, where first an initial solution is obtained by setting x j = 1 for all j ∈ N, and then iteratively choosing the variable i which can be changed from x i = 1 to x i = 0 with minimal loss in the objective function.…”

Section: Heuristicsmentioning

confidence: 99%

Upper bounds and exact algorithms for -dispersion problems

Pisinger

2006

Computers & Operations Research

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Section: Heuristicsmentioning

confidence: 99%

Upper bounds and exact algorithms for -dispersion problems

Pisinger

2006

Computers & Operations Research

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“…The number of non-zero coefficients of the objective function divided by n·(n+1)/2 is called as density (d) [3], which indicates the level of interaction between items and may have influence on the problem difficulty.…”

Section: The Quadratic Knapsack Problem (Qkp)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Classical exact methods [39] for solving the QKP are the branch and bound (B&B) algorithms, where numerous upper bounds have been obtained, by using techniques such as derivation of upper planes [20], Lagrangian relaxation [24], reformulation [10], linearization [3], Lagrangian decomposition [4,5,41], semidefinite relaxation [26], and reduction strategies [24,45], etc.…”

Section: Introductionmentioning

confidence: 99%

“…The knapsack problems [38] have been intensively studied due to both its theoretical interest and its wide practical applicability. Due to its generality, the QKP has more practically applications [3,20], moreover, several graph problems can be formulated as the QKP [4,45].The KP is NP-hard, although it can be solved exactly in pseudo-polynomial time through dynamic programming based on Bellman recursion [21]. As a generalized version, the QKP is much harder than the KP [10,45], where the graph structure associated with coefficients of objective function plays a important role in its solution [46].…”

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confidence: 99%

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A Mini-Swarm for the Quadratic Knapsack Problem

Xie

Liu

2007

2007 IEEE Swarm Intelligence Symposium

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Abstract-The 0-1 quadratic knapsack problem (QKP) is a hard computational problem, which is a generalization of the knapsack problem (KP). In this paper, a mini-Swarm system is presented. Each agent, realized with minor declarative knowledge and simple behavioral rules, searches on a structural landscape of the problem through the guided generate-and-test behavior under the law of socially biased individual learning, and cooperates with others by indirect interactions. The formal decomposition of behaviors allows understanding and reusing elemental operators, while utilizes the heuristic information on the landscape. The results on a collection of the QKP instances by mini-Swarm versions are compared with that of both a branch-and-bound algorithm and a greedy genetic algorithm, which show its effectiveness. I. INTRO DUCTIONThe 0-1 quadratic knapsack problem (QKP) was introduced by Gallo et al. [20], which consists in choosing elements from n items for maximizing a quadratic profit objective function subject to a linear capacity constraint. The QKP is a generalization of the 0-1 knapsack problem (KP) [38], by restricting all the quadratic coefficients to zero. The knapsack problems [38] have been intensively studied due to both its theoretical interest and its wide practical applicability. Due to its generality, the QKP has more practically applications [3,20], moreover, several graph problems can be formulated as the QKP [4,45].The KP is NP-hard, although it can be solved exactly in pseudo-polynomial time through dynamic programming based on Bellman recursion [21]. As a generalized version, the QKP is much harder than the KP [10,45], where the graph structure associated with coefficients of objective function plays a important role in its solution [46].Due to the NP-hardness in strong sense, the research efforts on solving the QKP may focus on exact methods, approximate algorithms and heuristic methods.Classical The studies on approximate algorithms, which seek guaranteed polynomial-time performance on approximate solution, were only limited on some special cases of the QKP, such as the classical KP [30], and the QKP where the underlying graph is edge series-parallel [46], etc.The heuristic methods arise from practical interest, which try to find good solution in high probability with acceptable computational efforts by considering certain imprecise knowledge on the structure of the problem. The linearization and exchange (LEX) heuristic [24] uses the best linear L2-approximation to build an associated linear KP and use it for determining items in a greedy way to form a good initial solution, which is then improved by an exchange method between certain items. Hua et al. [29] studied on the convex QKP by an approximate dynamic programming approach. Julstrom [32] also illustrated the capability of several heuristics, including greedy, genetic and greedy genetic algorithms on the QKP.Swarm algorithms [7,16,34] address on tackling a specified optimization task by multiple interacting agents, where each agent works u...

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